1. Excluding an induced wheel minor in graphs without large induced starsMujin Choi, Claire Hilaire, Martin Milanič, Sebastian Wiederrecht, 2026, published scientific conference contribution Abstract: We study a conjecture due to Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht stating that for any positive integer d and any planar graph H, the class of all K_{1,d}-free graphs without H as an induced minor has bounded tree-independence number. A k-wheel is the graph obtained from a cycle of length k by adding a vertex adjacent to all vertices of the cycle. We show that the conjecture of Dallard et al. is true when H is a k-wheel for any k at least 3. Our proof uses a generalization of the concept of brambles to tree-independence number. As a consequence of our main result, several important NP-hard problems such as Maximum Independent Set are tractable on K_{1,d}-free graphs without large induced wheel minors. Moreover, for fixed d and k, we provide a polynomial-time algorithm that, given a K_{1,d}-free graph G as input, finds an induced minor model of a k-wheel in G if one exists.
Keywords: induced minor, wheel, tree-independence number, Maximum Independent Set
Published in RUP: 25.03.2026; Views: 584; Downloads: 2
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2. Induced minor models : Structural properties and algorithmic consequencesNicolas Bousquet, Clément Jean Dallard, Maël Dumas, Claire Hilaire, Martin Milanič, Anthony Perez, Nicolas Trotignon, 2026, original scientific article Abstract: A graph H is an induced minor of G if there exists an induced minor model of H in G, that is, a collection of pairwise disjoint subsets of vertices of G labeled by the vertices of H, each inducing a connected subgraph in G, such that two vertices of H are adjacent if and only if there is an edge in G between the corresponding subsets. In this paper, we investigate structural properties of induced minor models, including bounds on treewidth and chromatic number of the subgraphs induced by minimal induced minor models. As algorithmic applications of our structural results, we make use of recent developments regarding tree-independence number to show that if H is the 4-wheel, the 5-vertex complete graph minus an edge, or a complete bipartite graph K2,q , then there is a polynomial-time algorithm to find in a given graph G an induced minor model of H in G, if there is one. We also develop an alternative polynomial-time algorithm for recognizing graphs that do not contain K2,3 as an induced minor, which revolves around the idea of detecting the induced subgraphs whose presence is forced when the input graph contains K2,3 as an induced minor. It turns out that all these induced subgraphs are Truemper configurations.
Keywords: induced minor, treewidth, chromatic number, tree-independence number, Truemper configuration
Published in RUP: 16.12.2025; Views: 883; Downloads: 6
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3. On relationships between treewidth, clique number, tree-independence number, and induced minors : master’s thesisÐorđe Vasić, 2025, master's thesis Keywords: treewidth, clique number, tree-independence number, induced minor, hered itary graph class, bipartite chain graph
Published in RUP: 11.09.2025; Views: 1414; Downloads: 23
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